Multiple-access Network Information-flow and Correction Codes

This work considers the multiple-access multicast error-correction scenario over a packetized network with $z$ malicious edge adversaries. The network has min-cut $m$ and packets of length $\mathrm{\ell }$, and each sink demands all information from the set of sources $\text{sources}$. The capacity region is characterized for both a “side-channel” model (where sources and sinks share some random bits that are secret from the adversary) and an “omniscient” adversarial model (where no limitations on the adversary’s knowledge are assumed). In the “side-channel” adversarial model, the use of a secret channel allows higher rates to be achieved compared to the “omniscient” adversarial model, and a polynomial-complexity capacity-achieving code is provided. For the “omniscient” adversarial model, two capacity-achieving constructions are given: the first is based on random subspace code design and has complexity exponential in $\ell m$,whilethesecondusesanovelmultiple-field-extensiontechniqueandhas$O(\ell m^{|\sources|})$ complexity, which is polynomial in the network size. Our code constructions are “end-to-end” in that all nodes except the sources and sinks are oblivious to the adversaries and may simply implement predesigned linear network codes (random or otherwise). Also, the sources act independently without knowledge of the data from other sources.